Quantum-Corrected Exponential Entropy Models

• Quantum-corrected exponential entropy models are a theoretical framework that extends the Bekenstein–Hawking area law by incorporating exponentially suppressed quantum corrections.
• These models emerge from diverse approaches such as loop quantum gravity, path-integral instantons, and holographic methods, influencing black hole thermodynamics and cosmological dynamics.
• The corrections yield novel thermodynamic relations, stabilize small black holes, and modify spacetime metrics, offering insights into nonperturbative quantum gravity effects.

Quantum-corrected models with exponential entropy constitute a broad theoretical framework in which the leading Bekenstein–Hawking area law receives non-perturbative, exponentially suppressed contributions. Such corrections play a central role in modern quantum gravity—arising from horizon microstate counting, path-integral instantons, and discrete quantum geometric structures. The ensuing modifications have profound consequences for black hole thermodynamics, critical phenomena in cosmological models, the structure of the semiclassical spacetime, and the statistical origin of gravitational entropy.

1. General Formulation of Exponential Entropy Corrections

In quantum-corrected models, the entropy $S$ associated with black hole horizons or cosmological apparent horizons is generalized to include an exponential (non-perturbative) term,

$$S(A) = S_{BH}(A) + \alpha\,e^{-\beta S_{BH}(A)} = \frac{A}{4} + \alpha\,e^{-\frac{A}{4}}$$

or equivalently (in Planck units, $A=4\pi r_+^2$ for spherical horizons),

$S(r_+) = \pi r_+^2 + \beta\,e^{-\alpha r_+^2}$

with dimensionless parameters $\alpha, \beta$ encoding theory-specific details (Anand, 22 Apr 2025, Sen et al., 2022, Pourhassan, 2020, Rivadeneira-Caro et al., 15 Sep 2025, He et al., 19 Sep 2025, Pourhassan et al., 2022). The exponential term is negligible for macroscopic areas ($A \gg 1$) but dominates in the quantum or Planck regime ($A \sim \mathcal{O}(1)$).

Such a structure is robust across multiple quantization frameworks:

• Loop quantum gravity and string-theoretic microstate counting generically produce these exponential corrections (including, for example, Kloosterman sums and discrete area spectra) (Anand, 22 Apr 2025, Sen et al., 2022).
• Path integral approaches and instanton effects yield analogous exponentially suppressed contributions in AdS/CFT and related settings (He et al., 19 Sep 2025).
• Cosmological models leverage the same form for the entropy associated to the apparent horizon, leading to significant changes in Friedmann equations and critical phenomena (Rivadeneira-Caro et al., 15 Sep 2025).

2. Universal Relations and Thermodynamic Structure

Quantum corrections modify the fundamental relations among thermodynamic quantities at the horizon. The generalized entropy function $S(r_h)$ enables a broad class of "quantum-corrected" laws. For a family of static, charged (A)dS black holes with horizon radius $r_h$, the following structures emerge:

1. First Law with Exponential Entropy:

$dM = T\,dS + \cdots$

where $T$ is fixed by the horizon surface gravity and $S$ contains exponential terms (Anand, 22 Apr 2025, Pourhassan, 2020).

1. Generalized Universal Relation: Extending the Goon–Penco relation, the response of the extremal mass $M_\mathrm{ext}$ to small deformations is controlled by the derivatives of the entropy:

$$-\,\frac{ \partial_\epsilon M_\mathrm{ext}}{ T\,\partial_\epsilon S }\Big|_{M_\mathrm{ext}} = \frac{\partial}{\partial S} S_{BH}( r_h(S) ) = 2\pi r_h(S)\,r_h'(S)$$

If the correction is turned off (pure area law), this reduces to unity, recovering the original Goon–Penco identity (Anand, 22 Apr 2025).

1. Partition Function and Thermodynamic Potentials: The quantum-corrected partition function factorizes as $Z = Z_0\,Z_c$, with the correction $Z_c$ governed by non-perturbative (Whittaker or instanton-type) sums. This enables closed-form computation of all thermodynamic potentials (mass, free energy, specific heat, enthalpy) in explicit models (Pourhassan, 2020, Pourhassan et al., 2022).
2. Equation of State and Smarr Relations: The inclusion of exponential entropy terms generically breaks the homogeneity required by the classical Smarr–Gibbs–Duhem relation, except for particular values of the black hole radius ($r_+$) or in limiting regimes (Pourhassan, 2020, Anand, 22 Apr 2025).

3. Black Hole Thermodynamics and Phase Structure

Thermodynamic quantities such as Hawking temperature, free energy, and specific heat acquire significant corrections at small area (Planck scale) due to the exponential term:

• Temperature and Specific Heat:

The corrected temperature takes the form (for the exponential correction)

$T = T_0 / \big(1 - \eta\,e^{-\beta S_0}\big)$

and the specific heat is rescaled accordingly, leading in certain models to a window of radii where small black holes become thermodynamically stable, signaling the emergence of Planck-scale remnants (Pourhassan, 2020, Pourhassan et al., 2022).

• Phase Transitions:
  • For the Born-Infeld BTZ black hole in massive gravity, a second spinodal (first-order-like) transition occurs due solely to the correction term (at $S_0=(1/\beta)\ln \alpha$, only when $\alpha>1$) (Pourhassan et al., 2022).
  • New first- or second-order transitions appear, as seen in curves of the specific heat and free energy, and a temperature-dependent virial expansion (with a novel virial coefficient) emerges in the equation of state.
  • In four-dimensional Schwarzschild and Schwarzschild–AdS black holes, the classically negative specific heat can become positive at small radii, stabilizing the black hole against evaporation and indicating a stable remnant phase (Pourhassan, 2020).

4. Spacetime Geometry and Quantum-Corrected Metrics

The exponential entropy term alters not only thermodynamics, but also the spacetime metric itself. Employing the first law in conjunction with the geometric definition of temperature, one reconstructs a unique quantum-modified metric: $$f(r) = 1 - \frac{2GM}{rc^2} + \frac{\ell_p^2}{2\pi r^2} e^{ -\frac{\pi r^2}{\ell_p^2} }$$ and

$$ds^2 = -f(r)c^2dt^2 + \frac{dr^2}{f(r)} + [r^2 - \frac{\ell_p^2}{\pi} e^{-\pi r^2 / \ell_p^2}]\,d\Omega^2$$

(Sen et al., 2022).

• Effective Matter Content: The modified Einstein tensor corresponds to a smeared energy density, with properties analogous to noncommutative-inspired Schwarzschild black holes.
• Causal Structure: The conformal structure and Penrose–Carter diagram remain Schwarzschild-like, but with horzion radius shifted inward by an exponentially small amount.
• Komar Energy and Smarr Relation: The Komar energy and quantum-corrected Smarr relation exhibit clear imprints of the exponential term, introducing Planck-scale corrections to observables such as the photon sphere and Hawking flux.

5. Quantum Microstates, Holography, and the Statistical Origin of Exponential Entropy

Quantum-corrected models with exponential entropy align naturally with the contemporary understanding of black hole microstates and holography.

• Hilbert Space Counting: In doubly-holographic models, the dimension of the black hole microstate Hilbert space is given by $N = \exp(S_q)$, where $S_q$ is the quantum-corrected thermodynamic entropy, which matches the generalised (quantum extremal surface) entropy,

$S_\text{gen} = S_{BH} + S_\text{CFT}$

The total Hilbert space for a two-sided eternal black hole is $\exp(2S_q)$ (He et al., 19 Sep 2025).

• Wormholes and Defect CFTs: The inclusion of $O(e^{-const N^2})$ corrections in AdS/CFT duality corresponds to brane- or D-instanton contributions, which modify extremality bounds and entropy in a universal fashion (Anand, 22 Apr 2025, He et al., 19 Sep 2025).
• Quantum Gravity Interpretation: Exponential corrections are interpreted as arising from tunneling sectors, discrete area spectra, or horizon-localized quantum states. They are generally invisible in perturbative expansions, encoding genuine nonperturbative information about quantum spacetime.

6. Cosmological Models and Exponential Entropy Corrections

In cosmology, exponential entropy corrections applied to the apparent horizon of FRLW universes yield modified Friedmann equations and rich dynamical phenomena:

• Modified Friedmann Dynamics:

$S(A) = \frac{A}{4} + \alpha e^{-A/4}$

$$-\frac{4\pi}{3}\rho = -\frac{1}{2R_{AH}^2} + \frac{\alpha}{2 R_{AH}^2} \left[ e^{-\pi R_{AH}^2} + \pi R_{AH}^2 \, \mathrm{Ei}(-\pi R_{AH}^2) \right]$$

$$-4\pi(\rho+p) = (1-\alpha e^{-\pi R_{AH}^2}) \dot{H}$$

(Rivadeneira-Caro et al., 15 Sep 2025).

• Phase Transition Structure: Depending on the sign and value of $\alpha$, the system exhibits
  • For $\alpha<0$: two positive-temperature critical points, first-order (swallowtail) transitions, and reentrant behavior,
  • For $\alpha>0$: one negative-temperature critical point, with no positive-$T$ transitions,
  • Phase structure controlled by transcendental equations for critical volume and temperature.
• Observational Constraints: Supernovae and cosmic chronometer data constrain $|\alpha| \lesssim 0.2$, and exponential corrections can mildly alleviate the $H_0$ tension by shifting $H(z)$ at moderate redshift.

7. Physical Implications and Applications

Exponential entropy corrections have far-reaching implications in quantum gravity and gravitational phenomenology:

• Remnants and Horizon Regularization: In quantum-corrected Schwarzschild solutions, the exponential correction leads to Planck-scale remnants and a natural regularization of curvature at the would-be singularity (Sen et al., 2022, Pourhassan, 2020).
• Phase Structure and Stability: New stable phases and multi-critical spinodal structure can appear even in purely gravitational black holes or cosmological models (Pourhassan et al., 2022, Rivadeneira-Caro et al., 15 Sep 2025).
• Weak Gravity Conjecture (WGC): The generalized universal relation ties the change in extremal mass caused by quantum corrections to entropy derivatives, supplying an "entropy-driven" proof of WGC in certain settings (Anand, 22 Apr 2025).
• Phenomenology: The thermodynamic modifications impact gravitational wave signatures (via photon sphere shifts), evaporation laws, and potentially cosmological observables.
• Model Building: The prescription for including exponential terms is robust: replace $S_{0}\to S_{0} + e^{-S_{0}}$ everywhere in thermodynamic and geometric formulae, ensuring consistency across both black hole and cosmological frameworks (Pourhassan, 2020).

Exponential entropy corrections constitute a universal feature of nonperturbative quantum gravity, with structural consequences for both gravitational thermodynamics and semiclassical spacetime geometry. Their emergence across microstate counts, holographic dualities, tunneling phenomena, and quantum geometry reinforces their centrality to the statistical and dynamical foundations of gravitational entropy.